Modified DOP

Although DOP is the most straightforward method of reflecting the geometry effect on positioning performance, it is not sufficient to reflect all details within the network, especially directional and system related bias. It might also be unfit to reflect non-ranging based network conditions, e.g. fingerprinting, which will not be discussed here. DOP is only used in this thesis to analyse the network conditions of ranging based networks when units are in LOS. When units are in NLOS, either the environment is separated into different LOS areas, or the influence on the ranging measurement between NLOS units will have to be considered, e.g. longer ranging measurement with lower accuracy between NLOS units.

The first factor that DOP cannot take into account is the accuracy of the ranging measurements between the rover and other units. However, the ranging accuracy directly influences the effectiveness of the ranging constraint in collaborative positioning while it is also one of the most influential factors on positioning accuracy. The states of the users within the collaborative network are constrained by the relative constraint, which is the ranging measurement plus an “error bound”. However, if this bound is set to a value smaller than the measurement error itself, i.e. the constraint is too “tight”, the state estimation would be pushed towards a wrong location. On the other hand, if the bound is much larger than the error, i.e.

constraint too “weak”, then it would not be able to sufficiently eliminate the observation noise and error. While the ranging measurements are used to calculate the DOP of each network, the quality of ranging would affect whether or not the DOP reflects the true geometry. Therefore, a modified DOP (MDOP ) which integrates the ranging quality is applied here to reflect the network geometry that is weighted by the measurement precision. The modified geometry matrix Amod is computed as below,

Amod =       ˆ xu−X1 a1·ˆr1 ˆ yu−Y1 a1·ˆr1 1 ˆ xu−X2 a2·ˆr2 ˆ yu−Y2 a2·ˆr2 1 ... ... ... ˆ xu−Xn an·ˆrn ˆ yu−Y1 an·ˆrn 1       (4.23)

where ai(i = 1, 2, ..., n) is a measurement accuracy coefficient derived from

RQI, hence a value between 0 and 1. Reliable measurements produce a closer to 1 and Amodwould be close to A. On the other hand, less reliable

measurements produce a closer to 0 and Amod would be much be larger

than the actual A. MDOP is computed from Amod as in Eq. 4.24, thus the

produced MDOP is usually larger than the original DOP.

M DOP = q

trace((AT

modAmod)−1) (4.24)

Another problem with applying just DOP is that the information on the relative “spread” of a network is condensed into a single value. However when observing the DOP equation, we can see that the DOP is the same when the product of the distance between the rover and the other two units are the same. This indicates that the same DOP can indicate two completely different networks where anchors are on different sides of the rover, as in Figure 4.33. Under normal circumstances, the two anchors in both situations will give the same restriction on the positioning precision in the diagonal direction along the line of the anchor and the rover (րւ) and unable to determine the position in the other diagonal direction (տց). Hence correctly indicated by the DOP value. However, in most ranging- based positioning scenarios, the ranging measurement are always positively biased, i.e. the ranging measurement is usually longer than the real distance due to disturbance in the propagation path. Given these conditions, the two scenarios in Figure 4.33 will no longer give the same restriction. Network 1 is able to constrain the error along the diagonal direction, but it is likely to be biased to one side of the rover due to the positive bias

4.5. Network geometry

in the ranging measurement (as indicated by the eclipse in Figure 4.33a). Yet network 2 can further constrain the error in both directions along the diagonal line as the ranging is coming from two different directions (error uncertainty is indicated by the eclipse in Figure 4.33b). Therefore, even with the same DOP, network constraints behave differently when the ranging measurement is known to be positively biased.

(a) Network 1 (b) Network 2

Figure 4.33: Examples of different network with same DOP (eclipse with dashed line indicates the error uncertainty in each network)

Moreover, when the two anchors are aligned, equation ∆x1 · ∆y2 =

∆x2· ∆y1 holds true. Hence the geometry matrix will be a singular matrix

where no valid DOP could be derived. Taking the example in Figure 4.33 one step further, there would be no valid DOP value if the two anchors in the first example are both located on the same corner , or if the anchors in the second example are located on the two corners but aligned with the rover. However as discussed, the network is able to provide relative constraint along the diagonal line. Therefore when the system detects an invalid DOP value, different situations are treated separately.

The third factor that DOP cannot reflect is the dynamic information during navigation, especially the directional information, e.g. the relative direction of the moving rover to the anchors as well as the rover system bias. Yet, this is also hard to detect if no prior knowledge is given. As an example, the simple simulation as shown in Figure 4.28 is applied here again. North direction is defined as upwards from the origin of the coordinate along the y-axis, East is defined as rightwards from the origin along the x-axis. Three scenarios are shown in Figure 4.34 where each network consists of two anchors at two of the selected locations. Scenario 1 is a network where the rover system bias is drifting northwards and

consists of anchor Tx1 and Tx4 (positioning results shown in magenta line). Scenario 2 shows a network of rover bias drifting southwards and anchors Tx1 and Tx4, and the bias of the rover is drifting downwards (shown in blue line). Scenario 3 shows a network of rover bias drifting northwards and anchor Tx2 and Tx3 (shown in cyan line).

Figure 4.34: Relative constraint effects on different system bias

Gyro drift is one of the largest error source in inertial measurements which pulls the rover offtrack from its original trajectory. This bias is almost always at an angle to the direction of the travelling trajectory and seldom follow the direction along the trajectory. Thus constraints along the direction of the actual bias direction are more useful in restricting the measurement error and preventing the bias from pulling the positioning estimation away from the truth. The relative constraint of two different networks and their effect on the measurement error in two directions are examined, where Network1 consists of Tx5, Tx6 and the rover, Network2 consists of Tx7, Tx8 and the rover, the location of each anchor is as shown in Figure 4.28. Figure 4.35 shows the cumulative distribution function (cdf) of the error distribution in both the East and North directions when applying the collaborative constraint from the two networks. As Tx5 and Tx6 are located on either side of the rover, the network constrains the error in the North direction better than the East direction, which is the travelling direction of the rover. Tx7 and Tx8 are located on either end of the travelling trajectory, thus constrains the error in the East direction better than the North direction.